L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.352 + 1.69i)3-s − 1.00i·4-s + (0.499 − 0.499i)5-s + (−1.44 − 0.949i)6-s + (−1.39 + 1.39i)7-s + (0.707 + 0.707i)8-s + (−2.75 + 1.19i)9-s + 0.705i·10-s + (3.39 + 3.39i)11-s + (1.69 − 0.352i)12-s − 1.97i·14-s + (1.02 + 0.670i)15-s − 1.00·16-s + 4.38·17-s + (1.09 − 2.79i)18-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.203 + 0.979i)3-s − 0.500i·4-s + (0.223 − 0.223i)5-s + (−0.591 − 0.387i)6-s + (−0.528 + 0.528i)7-s + (0.250 + 0.250i)8-s + (−0.916 + 0.398i)9-s + 0.223i·10-s + (1.02 + 1.02i)11-s + (0.489 − 0.101i)12-s − 0.528i·14-s + (0.263 + 0.173i)15-s − 0.250·16-s + 1.06·17-s + (0.259 − 0.657i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.107540449\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.107540449\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.352 - 1.69i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + (-0.499 + 0.499i)T - 5iT^{2} \) |
| 7 | \( 1 + (1.39 - 1.39i)T - 7iT^{2} \) |
| 11 | \( 1 + (-3.39 - 3.39i)T + 11iT^{2} \) |
| 17 | \( 1 - 4.38T + 17T^{2} \) |
| 19 | \( 1 + (-1.70 - 1.70i)T + 19iT^{2} \) |
| 23 | \( 1 + 0.998T + 23T^{2} \) |
| 29 | \( 1 - 0.998iT - 29T^{2} \) |
| 31 | \( 1 + (6.50 + 6.50i)T + 31iT^{2} \) |
| 37 | \( 1 + (4.10 - 4.10i)T - 37iT^{2} \) |
| 41 | \( 1 + (5.24 - 5.24i)T - 41iT^{2} \) |
| 43 | \( 1 + 8.88iT - 43T^{2} \) |
| 47 | \( 1 + (-0.352 - 0.352i)T + 47iT^{2} \) |
| 53 | \( 1 - 14.2iT - 53T^{2} \) |
| 59 | \( 1 + (0.998 + 0.998i)T + 59iT^{2} \) |
| 61 | \( 1 + 9.59T + 61T^{2} \) |
| 67 | \( 1 + (-5.79 - 5.79i)T + 67iT^{2} \) |
| 71 | \( 1 + (-7.13 + 7.13i)T - 71iT^{2} \) |
| 73 | \( 1 + (-1.70 + 1.70i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.207T + 79T^{2} \) |
| 83 | \( 1 + (9.17 - 9.17i)T - 83iT^{2} \) |
| 89 | \( 1 + (2.54 + 2.54i)T + 89iT^{2} \) |
| 97 | \( 1 + (-3.58 - 3.58i)T + 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.905482466576076221812485262598, −9.494235075735497065163843958234, −8.971854231076061485670979658130, −7.947488626192202195376604023054, −7.05334062899289403515148449644, −5.92862112457970132302165852925, −5.32620203064187754461274108165, −4.21597750149930053794152416122, −3.20009969777389487584469155601, −1.70247384135016876605941045469,
0.59654923725089217447263595513, 1.73457337819830743558394060559, 3.12046951350245076349840676552, 3.68894544549440768973101494018, 5.46035196203111570568671262708, 6.48921389250489345663235183745, 7.03555518375163123058243153369, 8.013294154396919530613604173837, 8.758372127834161312334473787828, 9.522310020187013604748759907311